2004 Fiscal Year Final Research Report Summary
Asymptotic analytical study of differential equations
| Project/Area Number |
15540186
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Sophia University |
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Principal Investigator |
UCHIYAMA Koichi Sophia University, Mathematics, Professor, 理工学部, 教授 (20053689)
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| Co-Investigator(Kenkyū-buntansha) |
OUCHI Sunao Sophia University, Mathematics, Professor, 理工学部, 教授 (00087082)
TAHARA Hidetoshi Sophia University, Mathematics, Professor, 理工学部, 教授 (60101028)
YOSHINO Kunio Sophia University, Mathematics, Lecturer, 理工学部, 講師 (60138378)
HIRATA Hitoshi Sophia University, Mathematics, Assistant, 理工学部, 助手 (20266076)
AOYAGI Miki Sophia University, Mathematics, Assistant, 理工学部, 助手 (90338434)
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| Project Period (FY) |
2003 – 2004
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| Keywords | asymptotic analysis / Borel summable / multi-summable / Fuchsian equation / totally characteristic / A-elliptic / distribution with support in a cone / stochastic complexity |
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| Research Abstract |
A. Asymptotic analysis of differential equations in complex domain.
Uchiyama showed that formal solutions to a certain nonlinear PDE with regular singularity converge by the method of majorant series. Ouchi showed Borel summability of formal solutions to a certain semilinear 1^<st> order singular PDE and its application to normal form of vector fields and multi-summability of formal solution to a certain PDE regarded as perturbation of ODE. Tahara obtained the structure of singularities of solutions to Fuchsian nonlinear PDE, necessary or sufficient conditions for existence of singular solutions to 1St order PDE's of normal type and (non)existence of singularities and uniqueness of solutions to nonlinear PDE of totally characteristic type.
B. Asymptotic analysis of differential equations in real domain and related analysis.
Uchiyama showed local uniquness of radial solutions to 2^<nd> order one dimensional p-elliptic equations and obtained with L.Paredes a candidate of nonlinear PDEs with regular singularity from one dimensinal model of 4^<th> order p-elliptic equations. Yoshino, with M. Suwa, characterized distribution of exponential type with applications and obtained results on positive definite distribution. Hirata gave a concrete representation of a family of solutions through elliptic functions to 3^<rd> order nonlinear PDE. Goto did reserch on asymptotic inclusion. Aoyagi gave asymptotic expasion of the stochastic comlexity of non-analytic learning machines.
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